Neither mission has a generally accepted special name in the way that San Diego is called “Mother of the Missions”. Our website lists the missions with these special names.
Santa Cruz is sometimes referred to as the “hard luck” mission because of its troubled history (early flooding that forced a reloation, the building of a pueblo (town) too close to the mission, the suspicious death of one of the padres, looting of the mission at the time of the pirate raid, the fact that it had the lowest population of all the missions etc.) To make matters worst the site was used to build a modern catholic church. All that remains today is a reduced size replica next door to this church, and (in an adjacent state park) one of the original Indian adobes, used for housing.
There was a priest who served at Santa Clara who was known for his piety and an uncanny ability to foretell events. Father Magin de Catala was known as The Prophet.
I hope these facts are of some help and stimulate your interest in the missions.